设Ai, Bi, Ci, Di 分别是蚂蚁爬行i米后所处A, B, C, D点的事件。

P(Ai)+P(Bi)+P(Ci)+P(Di)=1

根据对称性P(Bi)=P(Ci)=P(Di)

P(A1)=0, P(B1)=P(C1)=P(D1) = 1/3

P(A2)=P(A2|A1)P(A1)+P(A2|B1)P(B1)+P(A2|C1)P(C1)+P(A2|D1)P(D1)=

(1/3)P(B1)+(1/3)P(C1)+(1/3)P(D1)=P(B1)

P(A2)=P(B1)

P(A(i+1))=P(Bi)

P(A2)=1/3, P(B2)= (3-1)/3^2

P(A3)=(3-1)/3^2, P(B3)= (3^2-(3-1))/3^3

P(A4)= (3^2-(3-1))/3^3,  P(B4)= (3^3-3^2+3-1)/3^4

P(A5)= (3^3-3^2+3-1)/3^4, P(B5)= (3^4-3^3+3^2-3+1)/3^5

P(A6)= (3^4-3^3+3^2-3+1)/3^5, P(B6)= (3^5-3^4+3^3-3^2+3-1)/3^6

P(A7)= (3^5-3^4+3^3-3^2+3-1)/3^6=

(2(3^4)+2(3^2)+2)/3^6=

2(81+9+1)/3^6=

182/3^6

n=182